Module 5.5 · Lesson 5.5.1

Parametric Equations — Tracing Motion with a Parameter

Drive both x and y from a single parameter like time, so one dot can trace out any path — even ones an ordinary function could never draw.

Intuition

An ordinary function $y = f(x)$ can only assign one $y$ to each $x$ — it can never loop back or cross itself. But real motion doesn't work that way. A thrown ball's horizontal and vertical positions both depend on the same clock, time, and tracking that shared clock is exactly what a parameter does.

Core concept

In a parametric equation, both $x$ and $y$ are separate functions of a third variable $t$, the parameter. As $t$ ticks forward, the point $\big(x(t), y(t)\big)$ moves, tracing out a path. The parameter doesn't have to be time, but time is the most natural way to picture it: press play, and the dot moves.

t=2
Position functionsx(t)=10t, y(t)=20t−5t²
At t = 1(10, 15)
At t = 2(20, 20)
Landing: solve y(t)=0t = 4
Answerposition at t=2 is (20, 20)

Interactive Lab

Slide $t$ forward to watch the projectile's position trace its path. The trail behind the dot shows everywhere it has already been.

x y
t = 0.00 x(t) = 0.0 y(t) = 0.0

Key Idea

$$ \vec{r}(t) = \big\langle x(t),\, y(t) \big\rangle $$

A parametric curve is really a pair of ordinary functions of $t$, packaged together as a single position at each instant. Reading $x(t)$ and $y(t)$ separately tells you the horizontal and vertical motion; reading them together tells you where the point actually is on the plane.

$t$The parameter — often time, always increasing steadily
$x(t)$The horizontal position as a function of the parameter
$y(t)$The vertical position as a function of the parameter
$\vec{r}(t)$The combined position at parameter value $t$

Think Further

  1. Two different parametrizations could trace out the exact same visual path but reach the halfway point at different values of $t$. What would that difference actually represent physically?
  2. A circle can be written parametrically as $x(t)=\cos t$, $y(t)=\sin t$, but never as a single function $y=f(x)$. Why does the parameter make the loop possible when an ordinary function can't?
  3. If you solve $x(t)=10t$ for $t$ and substitute into $y(t)=20t-5t^2$, predict what kind of curve shape you'd get in terms of $x$ and $y$ alone.
Show suggested answers
  1. It would represent a difference in speed along the same path — one object moving faster or slower than the other while covering the identical geometric route.
  2. A function can only output one $y$ per $x$, but a circle needs two $y$ values for most $x$'s. Since $x(t)$ and $y(t)$ are tracked separately through $t$, the point can pass through the same $x$ twice at different times, which is exactly what a loop requires.
  3. Substituting $t = x/10$ gives $y = 20(x/10) - 5(x/10)^2 = 2x - 0.05x^2$, a downward-opening parabola in $x$ and $y$ — matching the projectile arc shape traced in the lab.